An easy completeness proof for the modal μ-calculus on finite trees

We give a complete axiomatization for the modal μ-calculus on finite trees. While the completeness of our axiomatization already follows from a more powerful result by Igor Walukiewicz in [11], our proof is easier and uses very different tools, inspired from model theory. We show that our approach generalizes to certain axiomatic extensions, and to the extension of the μ-calculus with graded modalities. We hope that the method might be helpful for other completeness proofs as well. The μ-calculus is an extension of modal logic with a fixpoint operator. In 1983, Dexter Kozen suggested an axiomatization and showed completeness for the aconjunctive fragment of the μ-calculus (see, e.g., [7]). It took more than ten years to prove completeness. This proof is due to Igor Walukiewicz [11] and is quite involved. It uses tableaux and the notion of disjunctive formula. We propose here a simpler proof in a particular case. More precisely, we prove the completeness of the Kozen axiomatization K extended with the axiom μx. x with respect to the class of finite tree models. Finite trees are a fundamental data structure in computer science, and logics on finite trees have received considerable attention in recent literature, motivated by applications in areas such as XML [1, 8]. Our argument consists of three steps. The first step consists of defining a notion of rank which plays the same role as the modal depth for modal formulas. One of the main properties of the rank is the following. In order to know whether a formula φ of rank n is true at a node w, it is enough to know which proposition letters are true at w and which formulas of rank at most n are true at the successor nodes of w. Another key property of the rank is that there are only finitely many formulas of a given rank (up to logical equivalence). The second step is to prove completeness of the μ-calculus with respect to generalized models, which are basically Kripke models augmented with a set of admissible subsets, in the style of Henkin semantics for second order logic. ⋆ The first author was supported by the Netherlands Organization for Scientific Research (NWO), under grant 639.021.508 and by ERC Advanced Grand Webdam on Foundation of Web data management. The second author is supported by VICI grant 639.073.501 of the NWO. We would like to thank Alexandru Baltag and Yde Venema for their comments on earlier drafts. Thanks also to the referees for helpful comments. The last step is inspired by the work of Kees Doets (see, e.g., [3]). Let us call a node in a generalized model n-good if there is a node in a finite tree model which satisfies exactly the same formulas of rank at most n. Using an induction principle, we show that every node in a generalized model satisfying μx. x is n-good. It is here that we use the main property of the rank. Finally, putting this together with the completeness for generalized models, we obtain completeness for the class of finite tree models. This argument can also be applied to some extensions of the logicK+μx. x. More precisely, we prove that when we add finitely many shallow axioms (as defined in [10]), we obtain a complete axiomatization for the corresponding class of finite trees. We also show that we can adapt our proof to show completeness for the graded μ-calculus extended with the axiom μx. x. Let us also mention that a similar method has been used for other completeness proofs in [6]. The paper is organized as follows. In section 1, we recall what is the Kozen axiomatization for the μ-calculus K and what is the intended semantics. In section 2, we define the notion of rank for a formula. In section 3, we give a definition for the generalized models and we show completeness of K with respect to the class of generalized models. In section 4, we use Kees Doets’ argument to obtain completeness of K + μx. x with respect to the class of finite tree models. In the last two sections, we give some examples of extensions ofK+μx. x to which we can apply our method in order to prove completeness. 1 Syntax, semantics and axiomatization We introduce the language and the Kripke semantics for the μ-calculus. We also recall the axiomatization given by Dexter Kozen. Definition 1. The μ-formulas over a set Prop of proposition letters and a set V ar of variables are given by φ ::= ⊤ | p | x | φ ∨ φ | ¬φ | ♦φ | μx.φ, where p ranges over the set Prop and x ranges over the set V ar of variables. In μx.φ, we require that the variable x appears only under an even number of negations in φ. We will assume that V ar is infinite. As usual, we let φ∧ψ, φ and νx.φ be abbreviations for ¬(¬φ∨¬ψ), ¬♦¬φ and ¬μx.¬[¬x/x]. The notions of subformula, bound variable, free variable and substitution are defined in the usual way. If φ and ψ are μ-formulas and if p is a proposition letter, we denote by φ[ψ/p] the formula obtained by replacing in φ each occurrence of p by ψ. Similarly, if x is a variable, we define φ[ψ/x]. A μ-sentence is a formula in which all the variables are bound. A μ-formula is a modal formula if it does not contain any subformula of the form μx.φ. Definition 2. A Kripke frame is a pair (W,R), whereW is a set and R a binary relation on W . A Kripke model is a triple (W,R, V ) where (W,R) is a Kripke frame and V : Prop → P(W ) a valuation. If (w, v) belongs to R, we say that w is a predecessor of v and v is a successor of w. Given a formula φ, a Kripke model M = (W,R, V ) and an assignment τ : V ar → P(W ), we define a subset [[φ]]M,τ that is interpreted as the set of points at which φ is true. The subset is defined by induction in the usual way. We only recall that [[μx.φ]]M,τ = ⋂ {U ⊆W : [[φ]]M,τ [x:=U ] ⊆ U}, where τ [x := U ] is the assignment τ ′ such that τ (x) = U and τ (y) = τ(y), for all y 6= x. Observe that the set [[μx.φ]]M,τ is the least fixpoint of the map φx : P(W ) → P(W ) defined by φx(U) := [[φ]]M,τ [x:=U ], for all U ⊆W . If w ∈ [[φ]]M,τ , we write M, w τ φ and we say that φ is true at w under the assignment τ . If φ is a sentence, we simply write M, w φ. A formula φ is true in M under an assignment τ if for all w ∈ W , we have M, w τ φ. In this case, we write M τ φ. A set Φ of formulas is true in a model M under an assignment τ , notation: M τ Φ, if for all φ in Φ, φ is true in M under τ . Finally, if (W,R) is a Kripke frame and for all valuations V and all assignments τ , φ is true in (W,R, V ) under the assignment τ , we say that φ is valid in (W,R) and we write (W,R) φ. Definition 3. The axiomatization of the Kozen system K consists of the following axioms and rules propositional tautologies, If ⊢ φ→ ψ and ⊢ φ, then ⊢ ψ (Modus ponens), If ⊢ φ, then ⊢ φ[p/ψ] (Substitution), ⊢ (p→ q) → ( p→ q) (K-axiom), If ⊢ φ, then ⊢ φ (Necessitation), ⊢ φ[x/μx.φ] → μx.φ (Fixpoint axiom), If ⊢ φ[x/ψ] → ψ, then ⊢ μx.φ→ ψ (Fixpoint rule), where x is not a bound variable of φ and no free variable of ψ is bound in φ. Definition 4. If Φ is a set of modal formulas, we write K+ Φ for the smallest set of modal formulas which contains the propositional tautologies, the K-axiom and is closed under the Modus Ponens, Substitution and Necessitation rules. We say that K + Φ is the extension of K by Φ. Note that if Φ is empty, we simply write K. Next, if Φ is a set of modal formulas, we denote by K+r Φ the smallest set of formulas which contains both K and Φ and is closed under the Modus Ponens and Necessitation rules. We call K+r Φ the restricted extension of K by Φ. Finally, if Φ is a set of μ-formulas, we write K + Φ for the smallest set of formulas which contains both K and Φ and is closed under the Modus Ponens, Substitution, Necessitation and Fixpoint rules. We say that K +Φ is the extension of K by Φ. Definition 5. Let (W,R) be a Kripke frame. A point r in W is a root if for all w inW , there is a sequence w0, . . . , wn such that w0 = r, wn = w and (wi, wi+1) belongs to R, for all i ∈ {0, . . . , n − 1}. The frame (W,R) is a tree if it has a root, every point distinct from the root has a unique predecessor and there is no sequence w0, . . . , wn+1 in W such that wn+1 = w0 and (wi, wi+1) belongs to R, for all i ∈ {0, . . . , n}. The frame (W,R) is a finite tree if it is a tree andW is finite. Finally, a finite tree Kripke model is a Kripke model (W,R, V ) such that (W,R) is a finite tree. Proposition 1. Let M = (W,R, V ) be a Kripke model. The formula μx. x is true at a point w in M iff there is no infinite sequence w0, w1 . . . in W such that w0 = w and (wi, wi+1) belongs to R, for all i ∈ N. In particular, the formula μx. x is true in M iff there is no infinite sequence w0, w1, . . . such that (wi, wi+1) belongs to R, for all i ∈ N. That is, iff M is conversely well-founded. We prove the completeness of the logic K + μx. x with respect to the class of finite tree Kripke models. That is, a formula φ is provable in K + μx. x iff it is valid in any finite tree Kripke model. In fact, this result can be derived from the completeness result proved by Igor Walukiewicz in [11]. We will give more details at the end of Section 4.